We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al.'99] also on proof size. [Alekhnovich and Razborov'03] established that if the clause-variable incidence graph of a conjunctive normal form (CNF) formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop their techniques to show that if one can “cluster” clauses and variables in a way that “respects the structure” of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. We also show how a weaker structural condition is sufficient to obtain lower bounds on width for the resolution proof system, and give a unified treatment that highlights similarities and differences between resolution and polynomial calculus (PC) lower bounds. As a corollary of our main technical theorem, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov'02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution (PCR). Thus, while onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis'93], both FPHP and onto-PHP formulas are hard even when restricted to bounded-degree expanders.

Pseudo-Boolean solvers hold out the theoretical potential of exponential improvements over conflict-driven clause learning (CDCL) SAT solvers, but in practice perform very poorly if the input is given in the standard conjunctive normal form (CNF) format. We present a technique to remedy this problem by recovering cardinality constraints from CNF on the fly during search. This is done by collecting potential building blocks of cardinality constraints during propagation and combining these blocks during conflict analysis. Our implementation has a non-negligible but manageable overhead when detection is not successful, and yields significant gains for some SAT competition and crafted benchmarks for which pseudo-Boolean reasoning is stronger than CDCL. It also boosts performance for some native pseudo-Boolean formulas where this approach helps to improve learned constraints. Copyright

Constraint programming solvers support rich global constraints and propagators, which make them both powerful and hard to debug. In the Boolean satisfiability community, proof-logging is the standard solution for generating trustworthy outputs, and this has become key to the social acceptability of computer-generated proofs. However, reusing this technology for constraint programming requires either much weaker propagation, or an impractical blowup in proof length. This paper demonstrates that simple, clean, and efficient proof logging is still possible for the all-different constraint, through pseudo-Boolean reasoning. We explain how such proofs can be expressed and verified mechanistically, describe an implementation, and discuss the broader implications for proof logging in constraint programming.

We show that there are CNF formulas which can be refuted in resolution in both small space and small width, but for which any small-width proof must have space exceeding by far the linear worst-case upper bound. This significantly strengthens the space-width trade-offs in [E. Ben-Sasson, SIAM J. Comput., 38 (2009), pp. 2511-2525], and provides one more example of trade-offs in the "supercritical" regime above worst case recently identified by [A.A. Razborov, J. ACM, 63 (2016), 16]. We obtain our results by using Razborov's new hardness condensation technique and combining it with the space lower bounds in [E. Ben-Sasson and J. Nordström, Short proofs may be spacious: An optimal separation of space and length in resolution, in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS '08), 2008, pp. 709-718].

Building on [Clegg et al.’96], [Impagliazzo et al.’99] established that if an unsatisfiable k-CNF formula over n variables has a refutation of size S in the polynomial calculus resolution proof system, then this formula also has a refutation of degree k + O(n log S). The proof of this works by converting a small-size refutation into a small-degree one, but at the expense of increasing the proof size exponentially. This raises the question of whether it is possible to achieve both small size and small degree in the same refutation, or whether the exponential blow-up is inherent. Using and extending ideas from [Thapen’16], who studied the analogous question for the resolution proof system, we prove that a strong size-degree trade-off is necessary.

We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if an only if there is a Nullstellensatz refutation of the pebbling formula over G in size t + 1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.

The conflict-driven clause learning (CDCL) paradigm has revolutionized SAT solving over the last two decades. Extending this approach to pseudo-Boolean (PB) solvers doing 0-1 linear programming holds the promise of further exponential improvements in theory, but intriguingly such gains have not materialized in practice. Also intriguingly, most PB extensions of CDCL use not the division rule in cutting planes as defined in [Cook et al.,'87] but instead the so-called saturation rule. To the best of our knowledge, there has been no study comparing the strengths of division and saturation in the context of conflict-driven PB learning, when all linear combinations of inequalities are required to cancel variables. We show that PB solvers with division instead of saturation can be exponentially stronger. In the other direction, we prove that simulating a single saturation step can require an exponential number of divisions. We also perform some experiments to see whether these phenomena can be observed in actual solvers. Our conclusion is that a careful combination of division and saturation seems to be crucial to harness more of the power of cutting planes

We prove that for k << (4)root n regular resolution requires length n(Omega(k)) to establish that an Erdos Renyi graph with appropriately chosen edge density does not contain a k-clique. This lower bound is optimal up to the multiplicative constant in the exponent, and also implies unconditional n(Omega(k)) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.

The last 20 years have seen dramatic improvements in the performance of algorithms for Boolean satisfiability-so-called SAT solvers-and today conflict-driven clause learning (CDCL) solvers are routinely used in a wide range of application areas. One serious short-coming of CDCL, however, is that the underlying method of reasoning is quite weak. A tantalizing solution is to instead use stronger pseudo-Boolean (PB) reasoning, but so far the promise of exponential gains in performance has failed to materialize-the increased theoretical strength seems hard to harness algorithmically, and in many applications CDCL-based methods are still superior. We propose a modified approach to pseudo-Boolean solving based on division instead of the saturation rule used in [Chai and Kuehlmann'05] and other PB solvers. In addition to resulting in a stronger conflict analysis, this also improves performance by keeping integer coefficient sizes down, and yields a very competitive solver as shown by the results in the Pseudo-Boolean Competitions 2015 and 2016.